Inductance

1. Introduction

This article is about components, as opposed to
Physics, therefore we will not dwell on the detail of magnetic theory, unless it
is useful in the design of real world inductors or there use in electronic
circuits. For example, familiar equations for induced voltage v based on a
time-changing magnetic flux Ø will be mentioned,

...(1)

Also, its relationship with inductance L
and an Alternating Current "AC"i is

...(2)

Providing
L is constant with respect to the current i.
This is not always the case, as materials used in the construction of inductors
can have a variable permeability with respect to magnetic flux, causing the
inductance to become current dependant. However applications influenced by this
complication are are usually treated with a less calculus oriented approach.

As
with many components, inductors come in all shapes and sizes. Some are designed
for radio frequency operation and have inductance values ranging from 1.5 nH to
well over 10 uH. Others are designed for power applications, and such
high current operation requires the effects of variable permeability to be
considered. Further inductor categories are designed for power supply decoupling
and the elimination of unwanted potential resonance's with associated decoupling
capacitors becomes an important issue.

This
article will describe some typical geometries involved in the construction of
typical inductors, and then display a number to common case styles.

2. Inductance Formula For Various Wire Geometries

2.1. Straight Length Of Cylindrical Wire

If we consider a round (cylindrical) with relative
permeability ur, length l, diameter d (and therefore radius r)
that is remote from any magnetic fields resulting from returning currents, i.e.

Note:

The permeability
of free space u0 , which is a scaling factor relating
current flow to magnetic flux ( ), has a defined value of u0 = 4 · pi · 10-7
A Wb-1 m-2. Magnetic
materials are those that exhibit higher permeabilityu than u0, so that u = ur · u0
where ur is referred to as the "relative
permeability". For many conducting materials u = 1
(copper, gold, silver) whilst some other conducting materials (iron, etc) have
u > 1. (Materials can be classified as paramagnetic, diamagnetic or
ferromagnetic, the latter category includes iron and its derivatives with
relative permeability ur up to 5,000 or so).

This length of wire will have a DC inductance predicted by,

...(3)

As mentioned, most common conductors have relative permeability ur
= 1 so equation (3) becomes

...(4)

This represents the inductance for a continuous current, i.e.
at DC. The actual value reduces slightly as the frequency increases, and falls
to an ultimate value at an infinite frequency predicted by,

...(5)

We note that the variable term inside the bracketed term does
not change greatly, namely from -¾ to -1. This transition is centered around a
frequency of 250 kHz, for a wire diameter of 1 mm. Although this may be
significant in switch-mode regulator circuits that operate at similar
frequencies, the ultimate value of inductance formula can be used at typical
Radio Frequencies (RF).

2.2. Two Parallel Cylindrical Wires

The previous example is somewhat hypothetical as it ignores
the effect of the return current path. If we consider an example of two parallel
wires, which might be speaker cable for example, the total inductance looking
into one end of the cable with the opposite ends joined together will produce a
different result from the simple sum of both paths, due to the mutual
interaction of their magnetic fields,

The new predicted inductance will now be,

...(6)

This assumes the length of the end joining wire segment is
small compared to the total cable length. To illustrate the effect of this
interaction, lets predict the inductance of 0.1 meters of copper wire with a
diameter of 1 mm. Its inductance, predicted from equation (5) will be 99.8 nH.
If we now form a cable of the same length, using the same wire, with say 5 mm
separation between the centers of the conductors, the new inductance predicted
by equation (6) will be only 94.1 nH ! Note that this is not only
less than the sum of each wire, it is even less than the original single wire
length's inductance by itself. We see, therefore the significance of mutual
coupling of magnetic fields, even in such a simple wire geometry as a 2-wire
cable.

2.2. Square Loop Of Cylindrical Wire

If the wire is bend into a square so that each side has a new
length l i.e. ¼ the total length of the wire,

Its self inductance
can be predicted from

...(7)

If we continue with the previous wire example, each side will
now be 25 mm long, so that l = 0.025 meters. The new inductance becomes L
= 63.16 nH. Once again, we see a reduction inductance from the original
L = 99.8 nH, due to interacting magnetic fields.

2.3. Circular Loop Of Cylindrical Wire

Now lets imagine a circular loop of wire,

Its inductance will
be predicted as,

...(8)

Providing r > 2 · d which would probably be
difficult manufacture otherwise. Returning to the same length of wire, we note
that the circumference of a circle is equal to 2 · pi · r, which in
this case would equal the length of the wire. This implies a circle radius of r
= 15.92 mm. Using equation (8) we now predict an inductance equal to L =
78.84 nH. This is slightly greater than the square loop inductance of L
= 63.16 nH, which can be explained from the maximally distant separation of
opposite sides of a circle compared the the situation enjoyed by a square. The
inductance of the circular loop is still less than the inductance of the same
wire length in free space however.

2.4. Solenoid Of Cylindrical Wire

So far we have looked at single straight wires, dual wire
cable and single wire loops. These geometries suit low values of inductance but
larger inductance values in a compact form rely on a multiple winding approach.
The simplest of these is the solenoid,
which consists of cascaded turns of wire round on a cylindrical former or left
"free standing" in air. The diameter D of this solenoid is
defined from wire center to opposite wire center, and the radius r is one
half this value. The minimum solenoid length is determined by the number of
turns N and the wire diameter d.

The actual solenoid can be "space round" so that
gaps are left between the windings, i.e.
. The inductance for solenoids that are long compared to their diameter
i.e. l >> D can be predicted as,

...(9)

This formula assumes that the magnetic flux is parallel to
the solenoid length, which occurs when the length is large compared to the
diameter. This may not always be practical however, as some inductors may have
few turns and be very much shorter than their diameter. In this case a modified
formula is needed,

...(10)

This correction factor K tends to 1 for long solenoids,
representing equation (9) and reduces as the radius to length ratio increases,
becoming about ½ for solenoids with a radius slightly larger than the length.
(The approximation coefficients were determined numerically from published data
up to ratios of 10, but appears smooth enough to extrapolate further)

2.5 Toroid Of Cylindrical Wire

The previous solenoid coil equation (10) is "exact"
for an infinitely long solenoid, in which the magnetic field runs parallel to
the length of the solenoid. Real world solenoids do not enjoy this situation,
and the magnetic field "fringes" at the ends tending towards a
circular shape.

If the solenoid is bent into a
torus, then it can appear to "act" as if it has infinite
length, as the enclosed magnetic field is parallel to the circumference of the
torus.

In the case of a circular torus, the total inductance is the
same as would be expected from an infinitely long solenoid, i.e.

...(11)

An important advantage of a toroidal inductor is that all the
magnetic field is enclosed, so that end leakage of magnetic field is avoided.
This property is very useful for reducing Electromagnetic Interference (EMI)
which is especially important in switch-mode regulator circuits containing large
circulating currents in close proximity to sensitive electronic components.

Several "practical" variations on the pure circular
toroid with circular cross section are common. The cross section area A
will often be rectangular, as opposed to circular. This has negligible effect on
the EMI characteristics, and equation (11) remains unmodified. In some cases,
for example Television High Voltage transformers required to generate 30 kV of
more for older vacuum-phosphor based viewing screens, Coupled inductors
comprising primary and secondary transformer circuits require a large physical
separation to prevent electrostatic breakdown, and construction difficulties
requires the use of a split core that can be screwed together after each winding
has been placed. Given this, a rectangular "toroid" is far easier to
fabricate that two semi-circular halves. The EMI properties are compromised to
some degree, but the arrangement is still superior to alternative geometries.

2.6 Adding Materials With Permeability > 1

So far we have only considered air wound inductors, or those
wound on non magnetic formers (ur = 1). If a ferromagnetic
former is used, for example iron, the new inductance will become,

...(12)

This equation assumes that all the magnetic flux is
"coupled" to the magnetic material, which will be quite accurate for
toroidal geometries, but this flux will be partially uncoupled for solenoids, as
the field will be "escaping" from each end. In this case, an effective
relative permeability ur, effective will be a better guide,
i.e.

...(13)

Obviously then 1 < ur, effective < ur.

3. Examples Of Typical Inductors

3.1. Inductors Used For Switch Mode Regulator Circuits

3.2. Inductors Used For Radio Frequency Circuits

These tend to be based on a solenoid topology, with either a
ceramic (u=1) former orferrite(u>>1) for larger inductance values (L > 1 uH). The SMD
case dimension range from 0402 to 1210 (0.04 inches long, 0.02 inches wide to
0.12 inches long, 0.10 inches wide).

Ferrite based inductors are generally best suited for
frequencies below 300 MHz, as ferrite losses decrease at lower frequencies. In
general, Intermediate frequencies, e.g. 70 MHz can use ferrite cored inductors
and show Quality "Q" values up to about 70 or so, while higher Radio
Frequencies (RF) tend to use a ceramic former.

3.3 Inductors Used For General Purpose Power Supply
Decoupling

Ferrite "beads" are often used for general purpose
power supply decoupling. These are based on a short length of wire surrounded by
ferrite. Different grades of ferrite are used to create impedance versus
frequency characteristics, almost always designed to have a high loss component,
(low Q) which prevents accidental resonance with additional decoupling capacitors.
Typical reactance values at 10~ 200 Ohms, across a wide range of frequencies,
and resistive and inductive components have similar magnitude.